Theorem xpen 5582

Description: Equinumerosity law for cross product. Proposition 4.22(b) of [Mendelson] p. 254.

Hypotheses

Ref	Expression
xpen.1	|- A e. _V
xpen.2	|- B e. _V
xpen.3	|- C e. _V
xpen.4	|- D e. _V

Assertion

Ref	Expression
xpen	|- ((A ~~ B /\ C ~~ D) -> (A X. C) ~~ (B X. D))

Proof of Theorem xpen

Step	Hyp	Ref	Expression
1		entr 5473	. 2 |- (((A X. C) ~~ (B X. C) /\ (B X. C) ~~ (B X. D)) -> (A X. C) ~~ (B X. D))
2		xpen.2	. . . . . 6 |- B e. _V
3		xpen.3	. . . . . 6 |- C e. _V
4	2, 3	xpdom2 5501	. . . . 5 |- (A ~<_ B -> (C X. A) ~<_ (C X. B))
5		xpen.1	. . . . . 6 |- A e. _V
6	5, 3	xpdom2 5501	. . . . 5 |- (B ~<_ A -> (C X. B) ~<_ (C X. A))
7	4, 6	anim12i 360	. . . 4 |- ((A ~<_ B /\ B ~<_ A) -> ((C X. A) ~<_ (C X. B) /\ (C X. B) ~<_ (C X. A)))
8		sbthbg 5521	. . . . 5 |- (B e. _V -> ((A ~<_ B /\ B ~<_ A) <-> A ~~ B))
9	2, 8	ax-mp 7	. . . 4 |- ((A ~<_ B /\ B ~<_ A) <-> A ~~ B)
10	3, 2	xpex 4096	. . . . 5 |- (C X. B) e. _V
11		sbthbg 5521	. . . . 5 |- ((C X. B) e. _V -> (((C X. A) ~<_ (C X. B) /\ (C X. B) ~<_ (C X. A)) <-> (C X. A) ~~ (C X. B)))
12	10, 11	ax-mp 7	. . . 4 |- (((C X. A) ~<_ (C X. B) /\ (C X. B) ~<_ (C X. A)) <-> (C X. A) ~~ (C X. B))
13	7, 9, 12	3imtr3i 235	. . 3 |- (A ~~ B -> (C X. A) ~~ (C X. B))
14	2, 3	xpex 4096	. . . . 5 |- (B X. C) e. _V
15	3, 2	xpcomen 5498	. . . . 5 |- (C X. B) ~~ (B X. C)
16		enen2 5541	. . . . 5 |- (((B X. C) e. _V /\ (C X. B) ~~ (B X. C)) -> ((C X. A) ~~ (C X. B) <-> (C X. A) ~~ (B X. C)))
17	14, 15, 16	mp2an 761	. . . 4 |- ((C X. A) ~~ (C X. B) <-> (C X. A) ~~ (B X. C))
18	5, 3	xpex 4096	. . . . 5 |- (A X. C) e. _V
19	3, 5	xpcomen 5498	. . . . 5 |- (C X. A) ~~ (A X. C)
20		enen1 5540	. . . . 5 |- (((A X. C) e. _V /\ (C X. A) ~~ (A X. C)) -> ((C X. A) ~~ (B X. C) <-> (A X. C) ~~ (B X. C)))
21	18, 19, 20	mp2an 761	. . . 4 |- ((C X. A) ~~ (B X. C) <-> (A X. C) ~~ (B X. C))
22	17, 21	bitri 190	. . 3 |- ((C X. A) ~~ (C X. B) <-> (A X. C) ~~ (B X. C))
23	13, 22	sylib 215	. 2 |- (A ~~ B -> (A X. C) ~~ (B X. C))
24		xpen.4	. . . . 5 |- D e. _V
25	24, 2	xpdom2 5501	. . . 4 |- (C ~<_ D -> (B X. C) ~<_ (B X. D))
26	3, 2	xpdom2 5501	. . . 4 |- (D ~<_ C -> (B X. D) ~<_ (B X. C))
27	25, 26	anim12i 360	. . 3 |- ((C ~<_ D /\ D ~<_ C) -> ((B X. C) ~<_ (B X. D) /\ (B X. D) ~<_ (B X. C)))
28		sbthbg 5521	. . . 4 |- (D e. _V -> ((C ~<_ D /\ D ~<_ C) <-> C ~~ D))
29	24, 28	ax-mp 7	. . 3 |- ((C ~<_ D /\ D ~<_ C) <-> C ~~ D)
30	2, 24	xpex 4096	. . . 4 |- (B X. D) e. _V
31		sbthbg 5521	. . . 4 |- ((B X. D) e. _V -> (((B X. C) ~<_ (B X. D) /\ (B X. D) ~<_ (B X. C)) <-> (B X. C) ~~ (B X. D)))
32	30, 31	ax-mp 7	. . 3 |- (((B X. C) ~<_ (B X. D) /\ (B X. D) ~<_ (B X. C)) <-> (B X. C) ~~ (B X. D))
33	27, 29, 32	3imtr3i 235	. 2 |- (C ~~ D -> (B X. C) ~~ (B X. D))
34	1, 23, 33	syl2an 503	1 |- ((A ~~ B /\ C ~~ D) -> (A X. C) ~~ (B X. D))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   e. wcel 1300  _Vcvv 2292   class class class wbr 3338   X. cxp 3984   ~~ cen 5423   ~<_ cdom 5424

This theorem is referenced by:  unxpdom2 5997  sucxpdom 5998  cdaassen 6080  mapcdaen 6082  xpomen 8769  qnnen 8772  infxpidmlem1 8821  infxpidmlem10 8830  infxpidmlem12 8832  xpeng 15691

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-rab 2112  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-er 5318  df-en 5427  df-dom 5428

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